Summary: On Parametric Polymorphism and Irrelevance in MartinL˜ of Type Theory
Project PI.R2, INRIA Rocquencourt and PPS, Paris
We devise a typed equality judgement for a predicative
version of Miquel's Implicit Calculus and complete it with
a calculus for explicit substitutions. The resulting theory
IITT, Implicit Intensional Type Theory, is shown consis
tent by a partial equivalence model. We further present a
bidirectional type checking and extraction algorithm and
briefly sketch the integration of another notion of irrele
vance, Awodey and Bauer's bracket types. This work is
aimed at providing a solid an practical foundation for ex
traction of efficient programs from type theory.
Keywords: Explicit Substitutions, Program Extraction,
Implicit Quantification, PER Model, Typed Equality.
Dependently typed programming languages such as
Agda  and Coq  allow the programmer to express